Question

If the equation $({\log }_{a}x{)}^2-{\log }_a(|x|{)}^2=3$ has repeated solution, then possible number of real values of $a$ is

Answer 1

$({\log }_{a}x{)}^2-{\log }_a(|x|{)}^2=3$
Clearly, $x>0$
Then, $({\log }_{a}x{)}^2-2{\log }_{a}x-3=0[\because x>0]$
$\Rightarrow ({\log }_{a}x-3)({\log }_{a}x+1)=0$
$\Rightarrow x=a^3\text{(or)}x=a^{-1}$
For repeated solution,
$a^3=a^{-1}\Rightarrow a=\pm 1$
But neither value of $a$ satisfies the given equation.
$\therefore$ No value of $a$ gives repeated solution for the given equation.